Convex optimization in the theory and practice of statistical estimation, prediction, and inference

这项拟议的研究详细阐述了申请人在过去十年的研究中获得的经验，集中在凸优化，特别是约束和不可微优化发挥关键作用的方法上。凸优化的影响不仅限于所考虑方法的算法方面，而且还超越了它们的理论--凸优化的数学，特别是对所调查公式的对偶版本的研究，以获得更多的理论见解。其中包括可行的推论策略，以及对其结果的概率评估。 从统计学的角度来看，统一的方面是关于收缩/正则化方法的研究，该方法在复合决策(经验贝叶斯)问题和“稀疏促进”正则化方法(如各种版本的基寻踪/套索、自适应套索、Dantzig选择器、非负绞线)中占有重要地位。提出的主题包括混合的非参数估计(能够处理维灾的复合决策问题的改进算法)；在经验贝叶斯预测中的附加形状约束方法(具有单峰限制的混合)；正则化周期图分析(也包括L1和分位周期图以及L1和L2惩罚)，以及在非等间隔时间序列数据中的周期搜索应用；在线性模型(设计实验分析中的分层组织的稀疏模型)、非参数回归(因变点考虑而产生的同一变量中的相加模型)以及分位数和合成分位数回归的主题中，出于统计考虑而同时使用L2和L1惩罚。

The proposed research elaborates on the experience the applicant gained in about past ten years of his research, concentrating on methods where convex, especially constrained and non-differentiable optimization plays a pivotal role. The impact of convex optimization is not limited only to the algorithmic aspects of the considered methods, but transcends also into their theory - the mathematics of convex optimization, in particular the study of the dual versions of the investigated formulations, is employed to obtain additional theoretical insights. These include viable strategies for the inference, the probabilistic evaluation of their results.  From the statistical point of view, the unifying aspect is the study of methods involving shrinkage/regularization, a prominent place occupied by compound decision (empirical Bayes) problems and "sparsity-promoting" regularization methods (like various versions of basis pursuit/lasso, adaptive lasso, Dantzig selector, nonnegative garrote). The proposed topics include nonparametric estimation of mixtures (improved algorithms for compound decision problems capable of dealing with the curse of dimensionality); additional shape constrained approaches in empirical Bayes prediction (mixtures with unimodality restrictions); regularized periodogram analysis (including also l1 and quantile periodograms and l1 and l2 penalties) with applications to period hunting in unequally spaced time series data; the simultaneous use of l2 and l1 penalties motivated by statistical considerations in linear models (hierarchically organized sparse models in the analysis of designed experiments); nonparametric regression (additive models in the same variable arising from change-point considerations); and topics in quantile and composite quantile regression.